Theorem cantnfres 9746

Description: The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)

Hypotheses

Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
cantnfrescl.d	⊢ (𝜑 → 𝐷 ∈ On)
cantnfrescl.b	⊢ (𝜑 → 𝐵 ⊆ 𝐷)
cantnfrescl.x	⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅)
cantnfrescl.a	⊢ (𝜑 → ∅ ∈ 𝐴)
cantnfrescl.t	⊢ 𝑇 = dom (𝐴 CNF 𝐷)
cantnfres.m	⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆)

Assertion

Ref	Expression
cantnfres	⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛 ∈ 𝐵 ↦ 𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛 ∈ 𝐷 ↦ 𝑋)))

Distinct variable groups:   𝐵,𝑛   𝐷,𝑛   𝐴,𝑛   𝜑,𝑛

Allowed substitution hints:   𝑆(𝑛)   𝑇(𝑛)   𝑋(𝑛)

Proof of Theorem cantnfres

Dummy variables 𝑘 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnfrescl.d	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ On)
2		cantnfrescl.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ⊆ 𝐷)
3		cantnfrescl.x	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅)
4	1, 2, 3	extmptsuppeq 8229	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) = ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))
5		oieq2 9582	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) = ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅) → OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) = OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)))
6	4, 5	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) = OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)))
7	6	fveq1d 6922	. . . . . . . . . 10 ⊢ (𝜑 → (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘) = (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))
8	7	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘) = (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))
9	8	oveq2d 7464	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) = (𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)))
10		suppssdm 8218	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) ⊆ dom (𝑛 ∈ 𝐵 ↦ 𝑋)
11		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝐵 ↦ 𝑋) = (𝑛 ∈ 𝐵 ↦ 𝑋)
12	11	dmmptss 6272	. . . . . . . . . . . . . 14 ⊢ dom (𝑛 ∈ 𝐵 ↦ 𝑋) ⊆ 𝐵
13	12	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝑛 ∈ 𝐵 ↦ 𝑋) ⊆ 𝐵)
14	10, 13	sstrid 4020	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) ⊆ 𝐵)
15	14	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) ⊆ 𝐵)
16		eqid 2740	. . . . . . . . . . . . . 14 ⊢ OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) = OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))
17	16	oif 9599	. . . . . . . . . . . . 13 ⊢ OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)):dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))⟶((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)
18	17	ffvelcdmi 7117	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) → (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘) ∈ ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))
19	18	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘) ∈ ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))
20	15, 19	sseldd 4009	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘) ∈ 𝐵)
21	20	fvresd 6940	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝑛 ∈ 𝐷 ↦ 𝑋) ↾ 𝐵)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) = ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)))
22	2	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → 𝐵 ⊆ 𝐷)
23	22	resmptd 6069	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛 ∈ 𝐷 ↦ 𝑋) ↾ 𝐵) = (𝑛 ∈ 𝐵 ↦ 𝑋))
24	23	fveq1d 6922	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝑛 ∈ 𝐷 ↦ 𝑋) ↾ 𝐵)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) = ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)))
25	8	fveq2d 6924	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) = ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)))
26	21, 24, 25	3eqtr3d 2788	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) = ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)))
27	9, 26	oveq12d 7466	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) = ((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))))
28	27	oveq1d 7463	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧) = (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧))
29	28	mpoeq3dva 7527	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)))
30	6	dmeqd 5930	. . . . . 6 ⊢ (𝜑 → dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) = dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)))
31		eqid 2740	. . . . . 6 ⊢ On = On
32		mpoeq12 7523	. . . . . 6 ⊢ ((dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) = dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)) ∧ On = On) → (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)))
33	30, 31, 32	sylancl 585	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)))
34	29, 33	eqtrd 2780	. . . 4 ⊢ (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)))
35		eqid 2740	. . . 4 ⊢ ∅ = ∅
36		seqomeq12 8510	. . . 4 ⊢ (((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) ∧ ∅ = ∅) → seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅))
37	34, 35, 36	sylancl 585	. . 3 ⊢ (𝜑 → seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅))
38	37, 30	fveq12d 6927	. 2 ⊢ (𝜑 → (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))))
39		cantnfs.s	. . 3 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
40		cantnfs.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
41		cantnfs.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ On)
42		cantnfres.m	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆)
43		eqid 2740	. . 3 ⊢ seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)
44	39, 40, 41, 16, 42, 43	cantnfval2 9738	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛 ∈ 𝐵 ↦ 𝑋)) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))))
45		cantnfrescl.t	. . 3 ⊢ 𝑇 = dom (𝐴 CNF 𝐷)
46		eqid 2740	. . 3 ⊢ OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)) = OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))
47		cantnfrescl.a	. . . . 5 ⊢ (𝜑 → ∅ ∈ 𝐴)
48	39, 40, 41, 1, 2, 3, 47, 45	cantnfrescl 9745	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇))
49	42, 48	mpbid 232	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇)
50		eqid 2740	. . 3 ⊢ seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)
51	45, 40, 1, 46, 49, 50	cantnfval2 9738	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐷)‘(𝑛 ∈ 𝐷 ↦ 𝑋)) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))))
52	38, 44, 51	3eqtr4d 2790	1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛 ∈ 𝐵 ↦ 𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛 ∈ 𝐷 ↦ 𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  ∅c0 4352   ↦ cmpt 5249   E cep 5598  dom cdm 5700   ↾ cres 5702  Oncon0 6395  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450   supp csupp 8201  seq_ωcseqom 8503   +_o coa 8519   ·_o comu 8520   ↑_o coe 8521  OrdIsocoi 9578   CNF ccnf 9730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-seqom 8504  df-1o 8522  df-oadd 8526  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-oi 9579  df-cnf 9731

This theorem is referenced by:  cantnf2  43287